1 Chapter 42 In the previous chapters we have looked at the quantum behavior of electrons in various potentials (quantum wells, atoms, etc) but have neglected what happens at the center of the atom, the nucleus. For the last 90 years, a principal goal of physics has been to work out the quantum physics of nuclei themselves. In that same period new applications ranging from radiation therapy in cancer treatment to detecting radon gas in basements have been developed. Nuclear Physics 42-
Transcript
1
Chapter 42
In the previous chapters we have looked at the quantum behavior of electrons in various potentials (quantum wells, atoms, etc) but have neglected what happens at the center of the atom, the nucleus.
For the last 90 years, a principal goal of physics has been to work out the quantum physics of nuclei themselves. In that same period new applications ranging from radiation therapy in cancer treatment to detecting radon gas in basements have been developed.
Nuclear Physics
42-
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Discovering the Nucleus
42-
Fig. 42-1 Fig. 42-2
Fig. 42-3Rutherford, Geiger, Marsden experiments 1911-1913→nucleus is small in size, massive, and positively chargedNot plum pudding (evenly distributed)!
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Some Nuclear Properties
42-
Mass Binding EnergyNuclide Z N A Stability (u) Spin (MeV/nucleon)
Total number of neutrons and protons, mass number: A
Some Nuclear Terminology
42-
(42-1)A Z N= +Protons and neutrons are called nucleons.
197 Au: 197, Au 79, 197 79 118A Z N A Z= → = = − = − =
Nuclides with same Z but different A are called isotopes, e.g., 173Au to 204Au
Radionuclides decay (or disintegrate) by emitting a particle, thereby transforming into a different nuclide
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Nuclidic Chart
Organizing the Nuclides
42-Fig. 42-4
unstable
stable
642-Fig. 42-5
Organizing the Nuclides, cont'd
Isobar: nuclides with same mass number
unstable
stable
742-
Nuclear Radii
-151 femtometer 1 fermi 1 fm 10 m (42-2)= = =
13
0
0
(42-3)1.2 fm
r r Ar
=≈
Eq. 42-3 does not apply to halo nuclides, neutron rich nuclides in which some neutrons form a large halo around a spherical core of protons.
For example 8Li+n→ 9Li, r increases 4%, but when 9Li+2n→ 11Li (halo nuclide), r increases 30%,
842-
Atomic Masses
-271 u 1.66053873 10 kg (42-4)= ×
(excess mass) (42-6)M A∆ = −
M is the actual mass of the atom in atomic mass units and A is the mass number for the nucleus.
2 931.494013 MeV/u (42-5)c =
The actual mass of a nucleus is not simply the sum of the masses of all its constituent nucleons. Energy (Q = -∆m c2, which is equivalent to mass) can be released or absorbed in nuclear reaction forming the nucleus.
942-
Nuclear Binding Energies
( ) ( )2 2- binding energy (42-7)beE mc Mc∆ = ∑If we could tear apart a nucleus into its separate nucleons, the work required would be ∆Ebe.
The mass M of the nucleus is less than the total mass of its individual nucleons Σm → nucleus has less energy Mc2 than all the separated nucleons Σ(mc2) →this energy difference (binding energy) favors the nucleons binding into a nucleus.
binding energy per nucleon
( ) binding energy per nucleon (42-8)beben
EEA
∆∆ =
∆Eben represents the average energy holding each nucleon into the nucleus.
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If weakly bound nuclei transform into more strongly bound nuclei, the total masscan be reduced and the mass energy of the final state is lower than the mass energy of the initial state. Where does the excess initial mass energy go?
Nuclear Binding Energies
42-Fig. 42-6
Fission: A nucleus with a larger mass (U, Pu) splits into nuclei with smaller total mass (larger binding energy). Energy is released, e.g., nuclear reactor, nuclear weapons.
Fusion: Two nuclei combine to form a single more tightly bound nucleus, e.g., H+H →He hydrogen bomb and the sun
fusion fission
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Energy levels of nucleons confined in a nucleus are quantized just as for electrons confined in an atom. Do you notice a major quantitative difference?
Nuclear Energy Levels
42-Fig. 42-7 Fig. 39-18
Nucleons in an 28Al nucleus Electrons in a H atom
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Many nuclides have an intrinsic nuclear magnetic moment, which leads to intrinsic nuclear angular momentum or spin. While nuclear angular momentum is similar in magnitude to angular momenta of atomic electrons, nuclear magnetic moments are much smaller than typical atomic magnetic moments.
Nuclear Spin and Magnetism
42-
Attractive, short-range strong force binds quarks together to form protons and neutrons. This force "spills over" to bind nucleons in nuclei, overcoming the repulsive Coulomb force between protons
The Nuclear Force
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As shown in Fig. 42-4, most known nuclides are unstable/radioactive.
Radioactive Decay
42-
There is absolutely no way to predict whether any given nucleus in a radioactive sample will be among the small number of nuclei thatdecay during the next second. All have the same chance.
(42-11)dN Ndt
λ− =
Nuclear decay rate dN/dt is proportional to the number N of nuclei that can decay
= (42-12)dN dtN
λ−
0 0
=-N t
N t
dN dtN
λ∫ ∫ ( )0 0ln ln - - (42-13)N N t tλ− =
00
ln - (let 0) (42-14)N t tN
λ= =-
0
tN eN
λ=
-0 (radioactive decay) (42-15)tN N e λ=
14
Radioactive decay rate R = - dN/dt:
Radioactive Decay, cont'd
42-
(42-17)R Nλ=
-0 (radioactive decay) (42-16)tR R e λ=
( )-0 =tdNR N e N t
dtλλ λ= − =
The total decay rate of one or more nuclides is called the activity, with SI units becquerel
1 becquerel 1 Bq 1 decay per second= =
An older unit for activity, the curie, is still commonly used
101 curie 1 Ci 3.7 10 Bq= = ×
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Two common time measures of how long any given type of radionuclides lasts.
Half-life T½ (1/2 of starting nuclides have decayed) and mean life τ (1/e of starting nuclides have decayed).
Radioactive Decay, cont'd
42-
1/ 2ln 2 ln 2 (42-18)T τλ
= =
( ) 1/ 211/ 2 0 02
TR T R R e λ−= = 1/ 2ln 2Tλ
=
( ) 0 01R R R ee
λττ −= =1τλ
=
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When a nucleus undergoes alpha decay, it transforms to a different nuclide by emitting an alpha particle (a helium nucleus, 4He).
Alpha Decay
238 234 4U Th He (42-22)→ +The alpha decay of 238U can occur spontaneously (without an external source of energy) since the mass of 238U is greater than the mass of the total decay products. Disintegration energy Q=-∆Mc2. T1/2 is 4.5x109 y. Why so long? Why don't all 238U decay immediately?
Fig. 42-9
StrongForce
CoulombForce
Radionuclide Q Half-Life
238U 4.25 MeV 4.5 x 109 y228U 6.81 MeV 9.1 min
Table 42-2 Two Alpha Emitters Compared
42-
Exponentially sensitive tunneling
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A nucleus that decays spontaneously by emitting an electron or a positron (positively charged particle with mass of an electron) is said to undergo beta decay.
Beta Decay
32 32 -1/ 2P S + ( 14.3 d) (42-24)e Tυ→ + =
ν is the symbol for a neutrino, a neutral particle with a very small mass.
Both charge and nucleon number are conserved in beta decay
In both alpha and beta decay, the same amount of energy is released in the decay of a particular radionuclide (governed by the mass difference between the initial and final states). In beta-minus (plus) decay the energy is shared between the electron (positron) and the neutrino, so the electron (positron) energy can range from 0 to Kmax= Q.
max (42-28)Q K=
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The Neutrino
42-
Fig. 42-11
•In 1930 Wolfgang Pauli predicted the existence of the neutrino to 1) explain the wide range of energies for electrons and positrons in beta decay and 2) the missing angular momentum in beta decay measurements.
•Neutrinos are hard to detect; the mean free path of an energetic neutrino in water is several thousand light years! Earth is almost completely transparent to them.
•Neutrinos first detected in laboratory by Reines and Cowan in 1953
•Sun emits large number of neutrinos from its core. Exploding stars (supernovas) emit strong neutrino bursts which have been detected on earth by elaborate detectors located deep underground.
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Radioactivity and the Nuclidic Chart
42-
Fig. 42-12
proton richneutron richn p +e υ−→ +
p n +e υ+→ +
Valley of nuclidesstability band Large A, decay into valley by
repeated alpha emission and fission
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If you know the half-life of a radionuclide, you can use the decay of that radionuclide as a clock to measure time intervals.
Age of rocks: 40K→40Ar with T1/2=1.25x109 y, ratio of 40K to 40Ar in a rock can be used to determine when the rock was formed (and the 40K in the rock started transforming into stable 40Ar). This type of technique is used to date the earth and moon with a maximum age of about 4.5x109 y.
Shorter time intervals (pre-historic and historic dating): 14C→12C with T1/2=5730 y, 14C is produced at constant rate in upper atmosphere. Living organisms absorb both 14C and 12C while alive, maintaining a constant ratio. Once dead, no more C is absorbed and the remaining 14C begins to decay into stable 12C. By measuring 14C to 12C in organic matter (bones, fossils, parchment) one can determine when the organism that produced the organic matter died. This type of technique is used to date artifacts ranging from the charcoal in ancient campfires to the Dead Sea Scrolls.
Radioactive Dating
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Radiation (cosmic rays, radioactive emission from elements in earth's crust, human activity/industry) can damage living tissue. There are two parts in evaluating the effect of radiation on living tissue.
Measuring Radiation Dosage
42-
1. Absorbed Dose. Measure of radiation dose (energy per unit mass) actually absorbed by a specific object (for example a patient's hand or chest). SI unit is the gray (Gy). Older, commonly used unit is the rad (radiation absorbed dose).
A whole body, short term gamma-ray dose of 3 Gy (300 rad) will cause death in 50% of the population exposed to it. Typical average dose from natural and human origin is only 2 mGy (0.2 rad) per year.
1 Gy 1 J/kg 100 rad (42-32)= =
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Measuring Radiation Dosage, cont'd
42-
2. Dose Equivalent. Although different types of radiation (gamma rays, neutrons, etc) may deliver same energy to the body, they do not have the same biological effect. Dose equivalent allows us to rescale the absorbed dose to reflect the damage that a particular type of radiation can cause. The scaling factor is the RBE (relative biological effectiveness).
Dose Equivalent = RBE x Absorbed Dose
For x-rays and electrons: RBE=1
For slow neutrons: RBE=5
For alpha particles: RBE=10
The SI unit for dose equivalent is the sievert (Sv). An earlier unit rem(roentgen equivalent man) is still commonly used.
The National Council on Radiation Protection recommends that no one should receive an equivalent dose greater than 5 mSv per year.
1 Sv 100 rem (42-33)=
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Collective Model: nucleus like a drop of liquid. It correlates many facts nuclearmasses and binding energies and helps to explain nuclear fission and other nuclear reactions.
When projectile a enters target nucleus X, they form an excited, quasi-stable intermediate compound nucleus C, which after ~10-16s (a long time by nuclear standards) decays into nuclear state Y by emitting particle b. Once in state Cthe nucleus "forgets" how it got there and hence the decay does not depend on how the nucleus reached state C.
Nuclear Models
42-Fig. 42-13
(42-34)X a C Y b+ → → +
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Independent Particle Model: Unlike collective model, where nucleons move around at random and bump into each other frequently, in this model nucleons remain in well-defined quantum states and hardly collide at all.
Nucleons obey Pauli exclusion principle; no two nucleons in the nucleus may occupy the same quantum mechanical state at the same time. Collisions minimized since nucleons can only scatter into unoccupied states.
In atoms, electrons form shells containing: 2, 10, 18, 36, 54, 86,… electrons
→ magic electron numbers
In a nucleus, nucleons form shells containing: 2, 8, 20, 28, 50, 82, 126…nucleons→magic nucleon numbers. Nuclei with proton number Z or neutron number N has one of these values has a special stability (like chemical stability of atoms with completely closed electronic shells).
Alpha particles 4He are doubly magic and have exceptional stability
Stripping off nucleons from closed shells requires a great deal of energy, while removing a nucleon that is already outside a closed shell is much easier.
For example 121O (Z=51), removing 51st proton (already outside closed shell) only requires 5.8 MeV, removing 50th proton (inside closed shell) requires 11 MeV.
This is analogous to removing electrons from closed atomic shells. Removing the first electron (outside a closed atomic shell) in Na requires 5 eV while removing a second electron (in a closed atomic shell) requires 22 eV.
Nuclear Models, cont'd
42-
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A Combined Model: consider a nucleus with a small number of neutrons or protons outside a core of closed shells.
Outside nucleons occupy quantized states in potential well formed by core (independent model), but also interact with the core, deforming it and setting up "tidal wave" motions of rotation and vibration within it (collective model).
